Proving $n! > n^2$ by mathematical induction

Question

I'm trying to prove that $n! > n^2$ for $n\geq 4$ by use of mathematical induction, but I get to the inductive step and get lost. But I'm struggling with the inductive step as expected.

Answer 1

$4!=24>16=4^2$, so the basic step holds. Then:

$$ (n+1)! = (n+1) n! \color{red}{>} (n+1) n^2 > (n+1)(n+1) = (n+1)^2 $$ and we are fine. We used the inductive hypothesis in $\color{red}{>}$.

Answer 2

HINT: multiplying $$n!>n^2$$ by $n+1>0$ we get $$(n+1)!>n^2(n+1)$$ and now you have to show that $$n^2(n+1)>(n+1)^2$$ which is easy.